Rotation of a Rigid Body (lecture 10)

center of mass: defn

the motionless point in a rotating object that every other point in the object undergoes circular motion around

formula for finding the coordinates for center of mass for a discrete mass

M = total mass

m₁, m_2, m₃… = each chunk has its own mass

x₁, x₂, x₃ = each chunk has its own x-coord

y₁, y₂, y₃ = each chunk has its own y-coord

the center of mass is the mass-weighted center of objec

formula for finding the coordinates for center of mass for a continuous object

• for a continuous, non-discrete object, the total mass is cut up into infinitesimal pieces → integrate x and y in terms of mass

centers of mass of common shapes

rotational kinetic energy

• the kinetic energy due to rotation is called rotation kinetic energy

moment of inertia I formula for discrete objects

important in the rotational kinetic energy equation

parallel axis theorem for moment of inertia

the theorem is useful for when you want to quickly compute the moment of inertia for an off-center axis of rotation

moment of inertia I formula for continuous objects

find how to express dm and r properly

• always define the axis first

• make sure that r is the perpendicular distance from the axis

• choose mass slices that will reflect the symmetry of the shape: thin rings for disks, thin linear slices for rods, etc

if the mass is distributed further away from the axis of rotation, then the object would have a _______ moment of inertia I

greater

moments of inertia of common shapes and objects

moments of inertia exist even when the object is ______

not actively rotating

• the moment of inertia is an intrinsic property of an object

• it’s about how mass is distributed in relation to an axis of rotation

when can you sum up moments of inertia of multiple objects?

1. you can add moments of inertia when the objects rotate about the same axis

• if the axis of rotation is the same for all objects

• if the axis is fixed

• each object is rigidly attached to each other

• example: a rod with two masses attached to it: I_total = I_rod + I_m1 + I_m2

2. you can add moments of inertia when the objects are all components that make up the same object

3. you can add moments of inertia after applying the parallel axis theorem

• a piece’s center of mass can be shifted via the theorem so that it satisfies the conditions of the first two scenarios

torque: defn

the rotational effect of a force; the measure of how effectively a force causes an object to rotate about an axis.

• a small force far from the pivot can create more torque than a large force close to the pivot

• torque depends on where you push and the direction you’re pushing

• torque is a vector, with the magnitude representing how strong the rotational push is and the direction representing which way the rotation tends to happen

the direction of torque is given by the ____________

right hand rule

• point fingers along r

• curl towards F

• thumb = direction of the torque vector

formula for magnitude of torque

• units N * m

• r = distance measured from the axis of rotation to the point of application of the force

• F = the force applied

• angle = the angle between r and F

formula for vector of torque

since torque is a cross product of F and r, the torque vector is perpendicular to both the F vector and the r vector

net torque on an object

newton’s second law for rotational motion

where the torque here is only the magnitude

torque formula for a particle in a rigid body

rigid-body equilibrium

the net torque on the body is 0 and the net force on the body is 0

• a rigid body is a shape that won’t deform

• if forces cancel each other out: no linear acceleration

• if torque cancel each other out: no angular acceleration

rolling motion: defn

• combination of rotational and translational motion

• the most simple rolling motion involves an object (circular) rotates about an axis while moving alon a straight line trajectory

rolling motion: what is the tangential velocity of a particle located at the very bottom of a circular object engaging in rolling motion?

the bottom point (P) is always at rest

rolling motion: what is the tangential velocity of a particle located at the very top of a circular object engaging in rolling motion?

kinetic energy of a rolling object

rolling motion formulas

if the cross products are linear, the constants _________

factor out

angular momentum: vector formula

direction is perpendicular to the plane of rotation via the right hand rule (point your fingers along r, curl your fingers in the dir of p, and the direction of your thumb is the dir of the angular momentum)

angular momentum: magnitude formula

where the angle is the angle between r and v

derivative of angular momentum

= net torque

angular momentum of a rigid body

law of conservation of angular momentum (very similar to regular momentum)

net torque is 0 since net torque = to the rate of change of the angular momentum, and since angular momentum remains constant, net torque is 0

the relationship between angular momentum and angular velocity

angular momentum: all formulas